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Arcsine laws and interval partitions derived from a stable subordinator. (English) Zbl 0769.60014
Let $$(T_ s: s\geq 0)$$ be a subordinator, i.e. an increasing process with independent increments, stable with index $$\alpha$$, $$0<\alpha<1$$, and let $$(B_ t: t\geq 0)$$ be a process whose zero set is the closed range of the subordinator. Define $$\Gamma_ +(u)$$ to be the time $$(B_ t)$$ is positive for $$0\leq t<u$$. It is shown that, for each $$u>0$$, the distributions of $$\Gamma_ +(u)/u$$ and $$\Gamma_ +(T_ s)/T_ s$$ are the same. This is a generalization of Lévy’s result in which $$(B_ t)$$ is a Brownian motion and $$(T_ s)$$ is the inverse of the continuous local time process associated with the set of zeros of $$(B_ t)$$. This identity in distribution is extended to a large class of functionals derived from the lengths and signs of excursions of $$(B_ t)$$ away from 0 for the classical Brownian motion setup, and similar identities are obtained in the general case of a process whose zero set is the range of a stable subordinator.

MSC:
 60E07 Infinitely divisible distributions; stable distributions 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60J99 Markov processes 60J55 Local time and additive functionals 60J65 Brownian motion
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